Subgroups — #24917650

Order 10624.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128324	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128175	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128025	C₂	1	1	yes	maximal, characteristic, complemented
4	#19127990	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127980	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127919	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127919	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127851	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127837	C₂	1	1	yes	maximal, characteristic
10	#19127812	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127804	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127786	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127783	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127759	C₂	1	1	yes	maximal, characteristic, complemented
15	#485×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×D₃₃₂	C₂²	1	1	yes	characteristic
17	C₂×#13050577	C₂²	1	1	yes	characteristic
18	C₂×#13050577	C₂²	1	1	yes	characteristic
19	C₂×D₆₆₄	C₂²	1	1	yes	characteristic
20	C₂×D₆₆₄	C₂²	1	1	yes	characteristic
21	#14581791	C₂²	1	1	yes	characteristic, complemented
22	#14581787	—	2	1	no	—
23	#14581781	C₂²	1	1	yes	characteristic, complemented
24	#14581778	—	2	1	no	—
25	#14581774	—	2	1	no	—
26	#14581772	—	2	1	no	—
27	#14581766	C₂²	1	1	yes	characteristic
28	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic, complemented
29	C₄×D₃₃₂	—	2	1	no	—
30	C₄×D₃₃₂	—	2	1	no	—
31	#14581754	C₂²	1	1	yes	characteristic
32	#14581754	C₂²	1	1	yes	characteristic
33	#14581754	C₂²	1	1	yes	characteristic
34	#14581754	C₂²	1	1	yes	characteristic
35	#14581754	C₂²	1	1	yes	characteristic, complemented
36	#14581751	C₂²	1	1	yes	characteristic
37	C₂²⋊C₄×D₈₃	—	2	1	no	—
38	C₂²⋊C₄×D₈₃	—	2	1	no	—
39	#14581725	C₂²	1	1	yes	characteristic, complemented
40	#14581723	C₂²	1	1	yes	characteristic, complemented
41	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
43	#14581714	C₂²	1	1	yes	characteristic
44	#14581714	C₂²	1	1	yes	characteristic
45	#14581712	C₂²	1	1	yes	characteristic
46	#14581712	C₂²	1	1	yes	characteristic
47	#14581712	C₂²	1	1	yes	characteristic, complemented
48	#14581712	C₂²	1	1	yes	characteristic, complemented
49	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
50	#14581702	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic, complemented
52	C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
53	#117×C₈₃	C₂²	1	1	yes	characteristic
54	#107×C₈₃	C₂²	1	1	yes	characteristic, complemented
55	#106×C₈₃	C₂²	1	1	yes	characteristic, complemented
56	#103×C₈₃	C₂²	1	1	yes	characteristic, complemented
57	#103×C₈₃	C₂²	1	1	yes	characteristic, complemented
58	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
59	C₂³×D₈₃	—	2	1	no	—
60	C₂×Dic₁₆₆	—	2	1	no	—
61	C₂×D₃₃₂	—	2	1	no	—
62	C₂×D₃₃₂	—	2	2	no	—
63	C₂×D₃₃₂	—	2	2	no	—
64	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
65	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
66	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₈₃	—	2	1	no	—
68	C₂×C₄×D₈₃	—	2	1	no	—
69	C₂×C₄×D₈₃	—	2	1	no	—
70	C₂×C₄×D₈₃	—	2	1	no	—
71	C₂×C₄×D₈₃	—	2	1	no	—
72	C₂×C₄×D₈₃	—	2	1	no	—
73	#13050577	—	2	2	no	—
74	#13050577	—	2	2	no	—
75	D₆₆₄	—	2	2	no	—
76	D₆₆₄	—	2	2	no	—
77	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₈₃	—	2	1	no	—
80	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
81	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
82	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
83	Dic₈₃⋊C₄	—	2	1	no	—
84	C₂²⋊Dic₈₃	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₈₃	D₄	1	1	yes	characteristic
86	(C₂×D₈₃)⋊C₄	—	2	1	no	—
87	(C₂×D₈₃)⋊C₄	—	2	1	no	—
88	(C₂×D₈₃)⋊C₄	—	2	1	no	—
89	(C₂×D₈₃)⋊C₄	—	2	1	no	—
90	(C₂×D₈₃)⋊C₄	—	2	1	no	—
91	(C₂×D₈₃)⋊C₄	—	2	1	no	—
92	(C₂×D₈₃)⋊C₄	—	2	1	no	—
93	(C₂×D₈₃)⋊C₄	—	2	1	no	—
94	C₄×Dic₈₃	—	2	1	no	—
95	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
96	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₃	—	2	1	no	—
102	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₈₃	—	2	1	no	—
106	C₂²⋊C₄×C₈₃	—	2	1	no	—
107	C₄²×C₈₃	—	2	1	no	—
108	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₈₃	—	2	1	no	—
110	C₂²×D₈₃	—	2	1	no	—
111	C₂²×D₈₃	—	2	4	no	—
112	C₂²×D₈₃	—	2	1	no	—
113	C₂²×D₈₃	—	8	1	no	—
114	C₂²×D₈₃	—	8	1	no	—
115	Q₈×C₈₃	—	2	1	no	—
116	Q₈×C₈₃	—	2	1	no	—
117	C₂×C₄×C₈₃	—	2	1	no	—
118	C₂×C₄×C₈₃	—	2	1	no	—
119	C₂×C₄×C₈₃	—	2	1	no	—
120	C₂×C₄×C₈₃	—	2	1	no	—
121	C₂×C₄×C₈₃	—	2	1	no	—
122	C₂×C₄×C₈₃	—	2	1	no	—
123	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₈₃	—	8	1	no	—
126	C₂×Dic₈₃	—	2	1	no	—
127	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
131	D₃₃₂	—	2	4	no	—
132	D₃₃₂	—	2	1	no	—
133	D₃₃₂	—	2	1	no	—
134	D₃₃₂	—	2	1	no	—
135	D₃₃₂	—	2	1	no	—
136	D₃₃₂	#41	1	2	yes	—
137	D₃₃₂	#41	1	2	yes	—
138	C₄×D₈₃	—	8	1	no	—
139	C₄×D₈₃	—	8	1	no	—
140	C₄×D₈₃	—	8	1	no	—
141	C₄×D₈₃	—	8	1	no	—
142	C₄×D₈₃	—	8	1	no	—
143	Dic₁₆₆	—	8	1	no	—
144	C₈×C₈₃	—	2	1	no	—
145	C₈×C₈₃	—	2	1	no	—
146	C₈₃⋊C₈	—	2	1	no	—
147	C₈₃⋊C₈	—	2	1	no	—
148	C₂²×C₈₃	—	2	1	no	—
149	C₂²×C₈₃	—	2	1	no	—
150	C₂²×C₈₃	—	2	1	no	—
151	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₈₃	—	2	2	no	—
153	C₂×D₈₃	—	2	2	no	—
154	C₂×D₈₃	—	4	1	no	—
155	C₂×D₈₃	—	4	1	no	—
156	C₂×D₈₃	—	4	1	no	—
157	C₂×D₈₃	—	4	1	no	—
158	C₂×D₈₃	—	4	1	no	—
159	C₂×D₈₃	—	4	1	no	—
160	C₂×D₈₃	—	4	1	no	—
161	C₂×D₈₃	—	8	4	no	—
162	C₄×C₈₃	—	2	1	no	—
163	C₄×C₈₃	—	4	1	no	—
164	C₄×C₈₃	—	4	1	no	—
165	C₄×C₈₃	—	4	1	no	—
166	C₄×C₈₃	—	4	1	no	—
167	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
168	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
169	Dic₈₃	—	2	1	no	—
170	Dic₈₃	—	2	1	no	—
171	Dic₈₃	—	2	1	no	—
172	Dic₈₃	—	2	1	no	—
173	Dic₈₃	—	4	1	no	—
174	C₂×C₈₃	—	4	1	no	—
175	C₂×C₈₃	#525	1	1	yes	characteristic
176	C₂×C₈₃	#576	1	1	yes	characteristic
177	C₂×C₈₃	#577	1	1	yes	characteristic
178	D₈₃	—	4	2	no	—
179	D₈₃	—	4	2	no	—
180	D₈₃	—	8	1	no	—
181	#3118	—	83	1	no	maximal, Sylow
182	C₈₃	#3118	1	1	yes	characteristic, complemented, Sylow
183	#542	—	83	1	no	—
184	#518	—	83	1	no	—
185	#485	D₈₃	1	1	yes	characteristic, complemented
186	#482	—	83	1	no	—
187	#476	—	83	1	no	—
188	#474	—	83	1	no	—
189	#463	—	83	1	no	—
190	#463	—	83	1	no	—
191	#450	—	83	1	no	—
192	#447	—	83	1	no	—
193	#442	—	83	1	no	—
194	#440	—	83	1	no	—
195	C₄×SD₁₆	—	83	1	no	—
196	C₄×D₈	—	83	1	no	—
197	#433	—	83	1	no	—
198	C₂²×D₄	—	83	1	no	—
199	C₂×SD₁₆	—	83	1	no	—
200	C₂×SD₁₆	—	83	1	no	—
201	C₂×D₈	—	83	1	no	—
202	C₂×D₈	—	83	1	no	—
203	C₄⋊Q₈	—	83	1	no	—
204	#126	—	166	1	no	—
205	#124	—	83	1	no	—
206	#124	—	166	1	no	—
207	#123	—	166	1	no	—
208	C₂²⋊Q₈	—	166	1	no	—
209	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
210	#121	—	83	1	no	—
211	C₄×Q₈	—	83	1	no	—
212	C₄×D₄	—	83	1	no	—
213	C₄×D₄	—	83	1	no	—
214	C₄×D₄	—	83	1	no	—
215	C₄×D₄	—	83	1	no	—
216	C₄×D₄	—	83	1	no	—
217	C₄×D₄	—	166	1	no	—
218	C₄×D₄	—	166	1	no	—
219	#117	—	83	1	no	—
220	#117	C₂×D₈₃	1	1	yes	characteristic
221	C₂×C₂²⋊C₄	—	166	1	no	—
222	C₂×C₂²⋊C₄	—	166	1	no	—
223	#107	—	83	1	no	—
224	#107	C₂×D₈₃	1	1	yes	characteristic, complemented
225	#106	—	83	1	no	—
226	#106	C₂×D₈₃	1	1	yes	characteristic, complemented
227	C₄⋊C₈	—	83	1	no	—
228	C₄⋊C₈	—	83	1	no	—
229	#103	C₂×D₈₃	1	1	yes	characteristic, complemented
230	#103	C₂×D₈₃	1	1	yes	characteristic, complemented
231	#102	—	83	1	no	—
232	#102	—	83	1	no	—
233	#102	—	83	1	no	—
234	#102	—	83	1	no	—
235	#102	—	83	1	no	—
236	#102	—	83	1	no	—
237	C₂²⋊C₈	—	83	1	no	—
238	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
239	#97	—	83	1	no	—
240	C₄×C₈	—	83	1	no	—
241	C₂⁴	—	166	1	no	—
242	C₂×Q₈	—	166	1	no	—
243	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
244	C₂×D₄	—	83	1	no	—
245	C₂×D₄	—	83	1	no	—
246	C₂×D₄	—	83	1	no	—
247	C₂×D₄	—	166	2	no	—
248	C₂×D₄	—	166	2	no	—
249	C₂×D₄	—	166	1	no	—
250	C₂×D₄	—	166	1	no	—
251	C₂²×C₄	—	166	1	no	—
252	C₂²×C₄	—	166	1	no	—
253	C₂²×C₄	—	166	1	no	—
254	C₂²×C₄	—	166	1	no	—
255	C₂²×C₄	—	166	1	no	—
256	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
257	SD₁₆	—	166	2	no	—
258	SD₁₆	—	166	2	no	—
259	D₈	—	166	2	no	—
260	D₈	—	166	2	no	—
261	C₂×C₈	—	83	1	no	—
262	C₂×C₈	—	83	1	no	—
263	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	83	1	no	—
267	C₄⋊C₄	—	83	1	no	—
268	C₄⋊C₄	—	83	1	no	—
269	C₄⋊C₄	—	166	1	no	—
270	C₄⋊C₄	—	166	1	no	—
271	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
272	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
273	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
274	C₂²⋊C₄	—	2	1	no	—
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	83	1	no	—
277	C₂²⋊C₄	—	83	1	no	—
278	C₂²⋊C₄	—	166	1	no	—
279	C₂²⋊C₄	—	166	1	no	—
280	C₂²⋊C₄	—	166	1	no	—
281	C₂²⋊C₄	—	166	1	no	—
282	C₂²⋊C₄	—	166	1	no	—
283	C₂²⋊C₄	—	166	1	no	—
284	C₂²⋊C₄	—	166	1	no	—
285	C₂²⋊C₄	—	166	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	83	1	no	—
288	C₄²	—	83	1	no	—
289	C₄²	—	166	1	no	—
290	C₂³	—	664	1	no	—
291	C₂³	—	664	1	no	—
292	C₂³	—	166	4	no	—
293	C₂³	—	166	1	no	—
294	C₂³	—	166	1	no	—
295	C₂³	—	166	1	no	—
296	C₂³	D₄×D₈₃	1	1	yes	characteristic
297	Q₈	—	2	1	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	664	1	no	—
300	D₄	—	83	2	no	—
301	D₄	—	83	2	no	—
302	D₄	—	664	1	no	—
303	D₄	—	166	4	no	—
304	D₄	—	166	1	no	—
305	D₄	—	166	1	no	—
306	D₄	—	166	1	no	—
307	D₄	—	166	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	83	1	no	—
315	C₂×C₄	—	83	1	no	—
316	C₂×C₄	—	83	1	no	—
317	C₂×C₄	—	83	1	no	—
318	C₂×C₄	—	664	1	no	—
319	C₂×C₄	—	664	1	no	—
320	C₂×C₄	—	664	1	no	—
321	C₂×C₄	—	664	1	no	—
322	C₂×C₄	—	664	1	no	—
323	C₂×C₄	—	166	1	no	—
324	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
325	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₈	—	166	1	no	—
329	C₈	—	166	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	332	1	no	—
334	C₂²	—	332	1	no	—
335	C₂²	—	332	1	no	—
336	C₂²	—	332	1	no	—
337	C₂²	—	332	1	no	—
338	C₂²	—	332	1	no	—
339	C₂²	—	332	1	no	—
340	C₂²	—	664	4	no	—
341	C₂²	—	166	2	no	—
342	C₂²	—	166	2	no	—
343	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	332	1	no	—
350	C₄	—	166	1	no	—
351	C₄	—	166	1	no	—
352	C₄	—	166	1	no	—
353	C₄	—	166	1	no	—
354	C₄	#41×D₈₃	1	1	yes	characteristic
355	C₄	#41×D₈₃	1	1	yes	characteristic
356	C₂	—	4	1	no	—
357	C₂	—	332	2	no	—
358	C₂	—	332	2	no	—
359	C₂	—	664	1	no	—
360	C₂	#19128209	1	1	yes	characteristic
361	C₂	#19128503	1	1	yes	characteristic
362	C₂	#19128510	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#485×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19127837	C₂
hypercenter	#485	D₈₃
hypocenter	C₈₃	#3118
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #19128324	C₂	semidirect
2. #19128175	C₂	semidirect
3. #19128025	C₂	semidirect
4. #19127990	C₂	semidirect
5. #19127980	C₂	semidirect
6. #19127919	C₂	semidirect
7. #19127919	C₂	semidirect
8. #19127851	C₂	semidirect
10. #19127812	C₂	semidirect
11. #19127804	C₂	semidirect
12. #19127786	C₂	semidirect
13. #19127783	C₂	semidirect
14. #19127759	C₂	semidirect
15. #485×C₈₃	C₂	semidirect
21. #14581791	C₂²	semidirect
23. #14581781	C₂²	semidirect
28. Q₈×Dic₈₃	C₂²	semidirect
35. #14581754	C₂²	semidirect
39. #14581725	C₂²	semidirect
40. #14581723	C₂²	semidirect
41. Dic₈₃⋊C₈	C₂²	semidirect
42. Dic₈₃⋊C₈	C₂²	semidirect
47. #14581712	C₂²	semidirect
48. #14581712	C₂²	semidirect
50. #14581702	C₂²	semidirect
51. C₈×Dic₈₃	C₂²	semidirect
54. #107×C₈₃	C₂²	semidirect
55. #106×C₈₃	C₂²	semidirect
56. #103×C₈₃	C₂²	semidirect
57. #103×C₈₃	C₂²	semidirect
182. C₈₃	#3118	semidirect
185. #485	D₈₃	semidirect
224. #107	C₂×D₈₃	semidirect
226. #106	C₂×D₈₃	semidirect
229. #103	C₂×D₈₃	semidirect
230. #103	C₂×D₈₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#19128324	2
#19128175	2
#19128025	2
#19127990	2
#19127980	2
#19127919	2
#19127851	2
#19127837	2
#19127812	2
#19127804	2
#19127786	2
#19127783	2
#19127759	2
#485×C₈₃	2
#3118	83